Distributions (pdf) - stat310

Discrete distributions
(sample space is a subset of the integers)
Discrete uniform(a, b) a, b ∈ Z, a < b
Equally likely events labelled with integers
from a to b.
PMF
Support
Mean
Variance
1/n n = b a +1
{a, a +1,...,b 1,b}
a + b
2
n 2 1
12
Bernoulli(p) p ∈ [0, 1]
An event with two outcomes:
1 = success, 0 = failure
PMF
Support
MGF
Mean
Variance
1 p x =0
{0, 1}
p x =1
1 p + pe t
p
p(1 p)
Binomial(n, p) n = 1, 2, 3, ...; p ∈ [0, 1]
The number of successes in n Bernoulli(p)
trials (a count)
PMF
✓ ◆
n
p
x
x (1 p)
Support
MGF
Mean
Variance
{0, 1,...,n}
(1 p + pe t ) n
np
np(1 p)
n x
Geometric(p) p ∈ [0, 1]
The number of successes in Bernoulli(p)
trials until one failure.
PMF
Support
MGF
Mean
Variance
(1 p)p x
{0, 1, 2,...}
1 p
1 pe t
p
1 p
p
(1 p) 2
Negative binomial(r, p) r = 1, 2, 3, ... p ∈ [0, 1]
The number of successes in Bernoulli(p)
trials until r failures.
PMF
✓
x + r
x
1
Support
MGF
Mean
Variance
{0, 1, 2,...}
✓
1 p
1 pet ◆r p
r
1 p
p
r
(1 p) 2
◆
(1 p) r p x
Poisson(λ) λ > 0
Count of events that have constant rate,
and are independent of one another.
PMF
Support
MGF
Mean
Variance
x e
x!
{0, 1, 2,...}
exp( (e t
1))

Continuous distributions
(sample space is an interval on the real line)
Uniform(a, b) a, b ∈ R
Likelihood of event proportional to it’s
length.
PDF
CDF
Support
Mean
Variance
1
b a
x a
b a
[a, b]
a + b
2
(b a) 2
12
Exponential(θ) θ > 0
Waiting time until a Poisson event with
average waiting time θ.
PDF
CDF
Support
MGF
Mean
Variance
1 e x/
1 e x/
[0, 1)
1
1 t
✓
✓ 2
Exponential(λ) λ > 0
Waiting time until a Poisson event with rate
λ.
PDF
Mean
1/
e x
Gamma(α, θ) α > 0, θ > 0
Waiting time for α Poisson events with
average waiting time θ.
PDF
CDF
Support
MGF
Mean
Variance
⇥
( ) x
1 e x/⇥
No closed form
[0, 1)
1
(1 t) ↵
↵✓
↵✓ 2
Gamma(α, β) β > 0, θ > 0
Waiting time for α Poisson events with rate
β.
PDF
Mean
⇥
( ) x
/⇥
1 e ⇥x
Normal(μ,σ 2 ) μ ∈ R, σ 2 > 0
PDF
CDF
Support
MGF
Mean
Variance
1
p 2 ⇥ e
(x)
( 1, 1)
exp(µt + 1
2
µ
2
(x µ)2
2 2
2 t 2 )